The new coupled method of spectral elements and modal solution for wave
propagation in 3D global earth models (Capdeville et al., 2000; 2001;
Chaljub et al., 2001) is well adapted to
assess the study of Seismic attenuation in the crust and the D''
structure by forward modeling approach. To perform these studies,
numerical developments, such as the possibility to have a layer of
spectral elements between two modal solutions, are required and have
been performed this year.

To provide a better
understanding of long period seismic wave attenuation in the upper
layers of the Earth by evaluating the respective contributions to
observed attenuation due to intrinsic attenuation (anelasticity) and
to scattering effects in the heterogeneous crust and uppermost mantle.
To investigate the observed discrepancy between quality factor (Q)
measured using a normal mode approach on the one hand, and a propagating
surface wave approach on the other. To determine whether this discrepancy
(on the order of 15% higher Q measured by modes compared to surface waves
at 200 sec) may be due to the influence of scattering.
To provide synthetic datasets in different synthetic Earth models
to calibrate and analyze the method used to invert real data and produce
3D attenuation map of the Earth.

We also plan to study the 3D structure of the D" layer,
the region of the Earth just above the
Core Mantle Boundary (CMB), by performing seismic wave propagation
simulations in realistic 3D D" models and comparing results with observed
waveforms of seismic waves sensitive to this region of the earth (e.g.
Sdiffracted). We hope to obtain new constraints on the character of 3D
structure in D".

The coupled method for wave propagation in global Earth
models is based upon the coupling between the spectral element method
and a modal solution method. The Earth is decomposed, depending on the
problem addressed, in two or three spherical shells, one shell with 3D
lateral heterogeneities and one or two shells with only
spherically symmetric heterogeneities (see Figure 26.1).

Figure 26.1:
Sandwich of spectral elements between two modal
solutions. and are the two coupling interfaces.

Depending on the problem, the
heterogeneous part can be mapped as the whole mantle, down to the
core-mantle boundary, or restricted to the upper mantle or to the crust.
In the 3D shell, the solution is sought in terms of a particular finite element
method, the spectral element method, which is based on a high order variational
formulation in space and a second-order explicit scheme in time. In the
spherically symmetric domain, the solution is sought in terms of a modal
solution in the frequency domain after expansion on the
spherical harmonics basis. The spectral element method combines the
geometrical flexibility of classical finite element method with the
exponential convergence rate associated with spectral techniques. It avoids pole
problems and allows local mesh refinement, using a non conforming
discretization (see Figure 26.2),

Figure 26.2:
Non conforming spectral element mesh used for the model
PREM. The coupling is performed a the Core Mantle Boundary.

for the resolution of sharp variations and topographical features along
interfaces. The modal solution is based on the spectral transform method and is
expressed in the spherical harmonics basis allowing an accurate and isotropic
representation in the spherically symmetric domain. The coupling between the
spectral element method, formulated in the space and time domain, and the modal
summation method, formulated in the frequency and wave-number domain,
requires some original solution. Within the spectral element method, the
coupling is introduced via a dynamic interface operator, a
Dirichlet-to-Neumann (DtN) operator. This operator can be explicitly
constructed in frequency and in generalized spherical harmonics. The back
transformation in the space and time domain requires however special attention
and an optimal asymptotic regularization. Such a coupling allows a significant
speed-up in the simulation of the wavefield propagation, and the computation of
synthetic seismograms in realistic earth models.

A description of the method with illustrations can be found on
http://seismo.berkeley.edu/yann

The development of the "sandwich" coupling. Six month ago, the only
configuration allowed was a spectral element outer shell coupled
with a modal solution inner sphere. To study the D" at high frequency,
a sandwich of spectral elements between two modal solutions is
required. This implies to develop the coupling of an outer shell
modal solution with a spectral element inner sphere, which has been performed
with success this year.

Thanks to Dimitri Komatitsch who gave us his attenuation code for the
spectral element method.
The computation were made using the computational resources of the
NERSC, especially the IBM SP, under repo mp342 starting project.